173 research outputs found
The 3D Spin Geometry of the Quantum Two-Sphere
We study a three-dimensional differential calculus on the standard Podles
quantum two-sphere S^2_q, coming from the Woronowicz 4D+ differential calculus
on the quantum group SU_q(2). We use a frame bundle approach to give an
explicit description of the space of forms on S^2_q and its associated spin
geometry in terms of a natural spectral triple over S^2_q. We equip this
spectral triple with a real structure for which the commutant property and the
first order condition are satisfied up to infinitesimals of arbitrary order.Comment: v2: 25 pages; minor change
Maximal quadratic modules on *-rings
We generalize the notion of and results on maximal proper quadratic modules
from commutative unital rings to -rings and discuss the relation of this
generalization to recent developments in noncommutative real algebraic
geometry. The simplest example of a maximal proper quadratic module is the cone
of all positive semidefinite complex matrices of a fixed dimension. We show
that the support of a maximal proper quadratic module is the symmetric part of
a prime -ideal, that every maximal proper quadratic module in a
Noetherian -ring comes from a maximal proper quadratic module in a simple
artinian ring with involution and that maximal proper quadratic modules satisfy
an intersection theorem. As an application we obtain the following extension of
Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let be an
element of the Weyl algebra which is not negative semidefinite
in the Schr\" odinger representation. It is shown that under some conditions
there exists an integer and elements such
that is a finite sum of hermitian squares. This
result is not a proper generalization however because we don't have the bound
.Comment: 11 page
Twisted Hochschild Homology of Quantum Hyperplanes
We calculate the Hochschild dimension of quantum hyperplanes using the
twisted Hochschild homology.Comment: 12 pages, LaTe
Metrics and Pairs of Left and Right Connections on Bimodules
Properties of metrics and pairs consisting of left and right connections are
studied on the bimodules of differential 1-forms. Those bimodules are obtained
from the derivation based calculus of an algebra of matrix valued functions,
and an SL\sb q(2,\IC)-covariant calculus of the quantum plane plane at a
generic and the cubic root of unity. It is shown that, in the
aforementioned examples, giving up the middle-linearity of metrics
significantly enlarges the space of metrics. A~metric compatibility condition
for the pairs of left and right connections is defined. Also, a compatibility
condition between a left and right connection is discussed. Consequences
entailed by reducing to the centre of a bimodule the domain of those conditions
are investigated in detail. Alternative ways of relating left and right
connections are considered.Comment: 16 pages, LaTeX, nofigure
The Noncommutative Geometry of the Quantum Projective Plane
We study the spectral geometry of the quantum projective plane CP^2_q, a
deformation of the complex projective plane CP^2, the simplest example of a
spin^c manifold which is not spin. In particular, we construct a Dirac operator
D which gives a 0^+ summable spectral triple, equivariant under U_q(su(3)). The
square of D is a central element for which left and right actions on spinors
coincide, a fact that is exploited to compute explicitly its spectrum.Comment: v2: 26 pages. Paper completely reorganized; no major change, several
minor one
All bicovariant differential calculi on Glq(3,C) and SLq(3,C)
All bicovariant first order differential calculi on the quantum group
GLq(3,C) are determined. There are two distinct one-parameter families of
calculi. In terms of a suitable basis of 1-forms the commutation relations can
be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce
bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For
generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3)
there are only two. The classical limit q-->1 of bicovariant calculi on
SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformation of
it which involves the Cartan-Killing metric.Comment: 24 pages, LaTe
A Hardy's Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra
In this article we consider linear operators satisfying a generalized
commutation relation of a type of the Heisenberg-Lie algebra. It is proven that
a generalized inequality of the Hardy's uncertainty principle lemma follows.
Its applications to time operators and abstract Dirac operators are also
investigated
Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime
We prove that the singularity structure of all n-point distributions of a
state of a generalised real free scalar field in curved spacetime can be
estimated if the two-point distribution is of Hadamard form. In particular this
applies to the real free scalar field and the result has applications in
perturbative quantum field theory, showing that the class of all Hadamard
states is the state space of interest. In our proof we assume that the field is
a generalised free field, i.e. that it satisies scalar (c-number) commutation
relations, but it need not satisfy an equation of motion. The same argument
also works for anti-commutation relations and it can be generalised to
vector-valued fields. To indicate the strengths and limitations of our
assumption we also prove the analogues of a theorem by Borchers and Zimmermann
on the self-adjointness of field operators and of a very weak form of the
Jost-Schroer theorem. The original proofs of these results in the Wightman
framework make use of analytic continuation arguments. In our case no
analyticity is assumed, but to some extent the scalar commutation relations can
take its place.Comment: 18 page
On Unbounded Composition Operators in -Spaces
Fundamental properties of unbounded composition operators in -spaces are
studied. Characterizations of normal and quasinormal composition operators are
provided. Formally normal composition operators are shown to be normal.
Composition operators generating Stieltjes moment sequences are completely
characterized. The unbounded counterparts of the celebrated Lambert's
characterizations of subnormality of bounded composition operators are shown to
be false. Various illustrative examples are supplied
Automorphisms of associative algebras and noncommutative geometry
A class of differential calculi is explored which is determined by a set of
automorphisms of the underlying associative algebra. Several examples are
presented. In particular, differential calculi on the quantum plane, the
-deformed plane and the quantum group GLpq(2) are recovered in this way.
Geometric structures like metrics and compatible linear connections are
introduced.Comment: 28 pages, some references added, several amendments of minor
importance, remark on modular group in section 8 omitted, to appear in J.
Phys.
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